PHYSICAL REVIEW E 93, 022111 (2016)
Universal behavior of crystalline membranes:
Crumpling transition and Poisson ratio of the flat phase
R. Cuerno,1 R. Gallardo Caballero,2 A. Gordillo-Guerrero,2,3 P. Monroy,4 and J. J. Ruiz-Lorenzo5,3
1
Departamento de Matemáticas and Grupo Interdisciplinar de Sistemas Complejos (GISC), Universidad Carlos III, 28911 Leganés, Spain
2
Departamento de Electrónica and Instituto de Computación Cientı́fica Avanzada (ICCAEx),
Universidad de Extremadura, 06071 Badajoz, Spain
3
Instituto de Biocomputación y Fı́sica de Sistemas Complejos (BIFI), 50018 Zaragoza, Spain
4
Instituto de Fı́sica Interdisciplinar y Sistemas Complejos IFISC (CSIC-UIB),
Campus Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain
5
Departamento de Fı́sica and Instituto de Computación Cientı́fica Avanzada (ICCAEx), Universidad de Extremadura, 06071 Badajoz, Spain
(Received 3 December 2015; published 8 February 2016)
We revisit the universal behavior of crystalline membranes at and below the crumpling transition, which
pertains to the mechanical properties of important soft and hard matter materials, such as the cytoskeleton
of red blood cells or graphene. Specifically, we perform large-scale Monte Carlo simulations of a triangulated
two-dimensional phantom network which is freely fluctuating in three-dimensional space. We obtain a continuous
crumpling transition characterized by critical exponents which we estimate accurately through the use of finite-size
techniques. By controlling the scaling corrections, we additionally compute with high accuracy the asymptotic
value of the Poisson ratio in the flat phase, thus characterizing the auxetic properties of this class of systems.
We obtain agreement with the value which is universally expected for polymerized membranes with a fixed
connectivity.
DOI: 10.1103/PhysRevE.93.022111
I. INTRODUCTION
Crystalline or polymerized membranes (CMs) constitute
a natural two-dimensional generalization of the simple idea
behind one-dimensional polymeric chains, namely, a twodimensional arrangement of monomers, which are connected
by rigid bonds that never break [1]. CMs are expected to
provide good approximations to the mechanical properties of
a number of interesting two-dimensional material systems.
Among others [2], they account for the thermal fluctuations
of the cytoskeleton of red blood cells [3,4] and provide an
accurate first step to describe the unique mechanical properties of graphene [5,6]. Such mechanical features include,
e.g., the existence of intrinsic, thermally induced ripples
in the graphene sheet [7,8], which can strongly influence
the electronic and magnetic properties [5] of this important
two-dimensional material.
In thermal equilibrium, the phase diagram of crystalline
membranes possesses a number of remarkable properties [9].
Thus, in contrast to polymers in solution [10], CMs are
flat (albeit rough) at low temperatures; namely, the local
normal directions at different points of the membrane have
a well-defined average orientation. Nevertheless, there is a
phase transition to a crumpled morphology at a finite value
of temperature, above which the local normal directions are
disordered. For the case of phantom CM, namely, in the
absence of self-avoidance, both theoretical and numerical
studies confirm this behavior [1,11,12], which also differs
from what is found for fluid membranes [13]. For selfavoiding membranes in physical dimensions, namely, a twodimensional network fluctuating in three-dimensional space,
there is no crumpling transition [14]. Rather, a unique
phase exists for all temperatures which is flat, with characteristics similar to the low-temperature phase of phantom
membranes.
2470-0045/2016/93(2)/022111(9)
The flat phase of crystalline membranes also hosts another
remarkable property [15], namely, auxetic behavior, which is
signaled by a negative Poisson ratio σ [16]. This property implies that the membrane expands transversely when stretched
longitudinally, contrary to experience with most common
elastic materials. Auxetic materials are expected to have good
mechanical properties [16], such as high energy absorption and
fracture resistance; see in particular [17] for a recent study of
the potential of graphene from this point of view, as assessed by
molecular dynamics simulations. Moreover, the conjectured
negative value of σ seems to be independent of the occurrence of self-avoidance constraints, σ = −1/3, as obtained
within the so-called self-consistent screening approximation
(SCSA) [18], having been put as characteristic of a unique
universality class for fixed-connectivity membranes [19].
In spite of these interesting properties of the flat phase of
crystalline membranes, they remain to be fully understood.
For instance, the nature of the crumpling transition is still
a subject of debate. While the computational and analytical
works quoted above mostly suggested that it is a continuous
transition, more recent results suggest, rather, a first-order
transition. Numerical results supporting the latter conclusion
have been obtained for a number of CM-related models in,
e.g., Ref. [20] (with a truncated Lennard-Jones potential) and
Refs. [21–23] (spherical topology). Results from nonperturbative renormalization group (RG) calculations are in agreement
with these [24]. Nevertheless, analogous RG studies [8,25,26]
seem to still favor a continuous, or perhaps weak first-order,
crumpling transition.
In this paper we revisit Monte Carlo simulations of
a discrete model of two-dimensional phantom membranes
fluctuating in three dimensions [2]. By implementing our
simulations in graphics processor units (GPUs), we are able to
reach large system sizes; we further implement an enhanced
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R. CUERNO et al.
PHYSICAL REVIEW E 93, 022111 (2016)
statistical analysis of the data. Our results favor the view
of the crumpling transition as a continuous one and feature
clear-cut convergence to the expected −1/3 value of the
Poisson ratio characterizing the universal auxetic properties
of the flat phase [18].
This paper is organized as follows. In Sec. II we recall the
basics of the continuum description of phantom membranes,
together with a detailed connection to the discrete model to
be simulated numerically. Section III details the observables
to be evaluated and their finite-size analysis. Our simulation
and statistical analysis methods are described in Sec. IV,
after which numerical results are presented in Sec. V. We
discuss our main results and summarize our conclusions in
Sec. VI. Three Appendices are provided for details on some
elasticity equations, RG estimates for the correction-to-scaling
exponent, and the practical implementation of our simulation
code in GPUs.
II. MODEL
A. Continuum description
For the sake of later reference, we briefly recall here the
basics of the Landau description of (phantom) polymerized
membranes in physical dimensions. Thus, a two-dimensional
membrane fluctuating in three-dimensional space can be
described geometrically as a vector field r(x 1 ,x 2 ) ∈ R3 , with
x = (x 1 ,x 2 ) ∈ R2 . The tangent vectors are tα = ∂α r, where,
as for all additional Greek indices in this work, α = 1,2.
These vectors allow us to define the metric tensor in the usual
way [27],
√
gαβ ≡ ∂α r · ∂β r ,
(1)
so that d s = d x g.
The most general form of the Landau free energy can
be written using general principles [11,28], namely, locality
and translational invariance (implying dependence only on the
local values of the tangent vectors tα and their derivatives),
rotational symmetry in R3 (implying dependence on scalar
products of the tangent vectors and their derivatives and
on even powers of these), and translational and rotational
invariance in R2 . Thus, one can write [11,28]
t
F [tα (x),T ] = d 2 s (tα )2 + u(tα tβ )2
2
α 2 κ
2
(2)
+ v tα t + (∂α tα ) ,
2
2
2
Taking into account the in-plane (u) and out-of-plane (h)
displacements and neglecting quadratic terms in uα , we can
write
uαβ =
1
(∂α uβ + ∂β uα + ∂α h∂β h).
2
(5)
The so-called elastic part of the Landau free energy (2),
namely,
t
FE [tα (x),T ] = d 2 s (tα )2 + u(tα tβ )2 + v(tα tα )2 , (6)
2
can now be written as
λ 2
FE (r ) = d 2 xfE (r ) = d 2 x μuαβ uαβ + uαα
, (7)
2
where λ and μ are the Lamé coefficients [29]. The remaining
part of the Landau free energy, namely,
κ
FC [tα (x),T ] = d 2 s (∂α tα )2
(8)
2
is a curvature contribution that can be written as
FC [tα (x),T ] = 2κ d 2 xKαβ Kβα ,
(9)
where indeed Kαβ is the extrinsic curvature [30].
B. Discrete model
We define our computational model on an N = L × L twodimensional triangular lattice like the one shown in Fig. 1,
embedded in three-dimensional space (see Fig. 2). The position
of the points is thus labeled by a three-dimensional vector r.
We also define triangular plaquettes with associated normal
vectors denoted as n. In general, we will denote points using
the indices i, j , k, etc., and plaquettes using a, b, c, etc.
The Hamiltonian we study is
H = HE + κHC ,
(10)
where self-avoidance is neglected [1].
One can parametrize the three-dimensional coordinates r
of points on a flat membrane taking as a reference the base
plane, namely, the plane determined by the average position of
all the atoms. Denoting the equilibrium position on this plane
by x and in-plane and perpendicular fluctuations by u and h,
respectively, we have
r(x) = [x + u(x),h(x)],
(3)
where (u(x),h(x)) = 0.
Moreover, on a flat membrane the strain tensor can be
written as [11]
uαβ = 12 (∂α r · ∂β r − δαβ ) = 12 (gαβ − δαβ ) .
(4)
FIG. 1. Lattice that has been employed in this work, for L = 6, in
which a four-color partition is illustrated. Points with the same color
can be updated simultaneously. This partition has been used to speed
up, in a highly parallel way, our numerical simulations using GPU
processors.
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PHYSICAL REVIEW E 93, 022111 (2016)
with the scaling law for ξ being
ξ (κ) ∼ |κ − κc |−ν .
(19)
Right at the critical point, the maximum of the specific heat
scales as
Cmax ∝ Ca + BLα/ν ,
FIG. 2. Sample membrane configuration at κ = 2 in the flat phase
for L = 24.
which includes an elastic part HE , given by
1
|ri − rj |2 ,
HE =
2 ij and a curvature contribution HC , which reads
1
HC =
|
na − nb |2 .
2 ab
(11)
where Ca describes the contribution from the analytical part
of the free energy.
We can further describe the space configuration of the membrane by considering the gyration radius for the distribution of
surface nodes, which is defined as
1 2
Rg =
Ri · R i ,
(21)
3N i
where Ri ≡ ri − rCM , with rCM being the position of the center
of mass of the surface. In addition, linear response theory
allows us to compute its κ derivative,
dRg2
= EC Rg2 − EC Rg2 ≡ EC Rg2 c .
(22)
dκ
Neglecting scaling corrections, the gyration radius scales with
the system size as
Rg ∼ LνF f (L1/ν (κ − κc )),
(12)
As usual, ij denotes nearest-neighbor points, while ab
denotes nearest-neighbor plaquettes.
Within a mean-field approach, it is possible to
show [2,31,32] that the above Hamiltonians retrieve the
corresponding contributions to the Landau free energy in the
continuum limit, namely,
1 α β
2
αβ
HE → d s uα uβ + uαβ u
,
(13)
2
(14)
HC → d 2 sKαβ Kβα .
(20)
(23)
which defines the Flory exponent νF . This exponent is related
to the Hausdorff dimension dH of the membrane by means of
νF =
2
.
dH
(24)
Near the crumpling transition, one finds dH = −4/η, where
η is the anomalous dimension of the r field [34]. In the flat
phase one has dH = 2 and νF = 1, while the high temperature
rough phase features νF = 0 and dH = ∞, which is linked to
2
a logarithmic divergence of the gyration radius, RG
∼ log L.
Therefore, the κ derivative of the gyration radius diverges at
the critical point κ = κc as
dRg2
III. OBSERVABLES AND THE
FINITE-SIZE SCALING METHOD
The specific heat can be computed as [33]
CV =
κ 2 2 EC − EC 2 ,
N
where
EC =
(15)
∝ L2νF +1/ν ,
(25)
dκ
an equation that will allow us to compute numerically the
Flory exponent. Note that, using the SCSA approximation, it
has been found that ν = 0.73 and hence dH = 2.74 [18].
Finally, we can estimate the Poisson ratio via (see Appendix A)
σ =
na · nb
(16)
<ab>
is the curvature energy. The behavior of the specific heat
near the crumpling transition that takes place at κ = κc is
characterized as
CV ∼ |κ − κc |−α .
n(x) · n(0) ∝ e−|x|/ξ ,
(18)
(26)
where g is the induced metric that can be estimated using the
vector which connects nearest-neighbor points, as described
in Appendix A. Note that we have assumed isotropy in the last
2
2
c = g22
c .
two terms in Eq. (26), specifically, g11
IV. NUMERICAL SIMULATIONS
(17)
The correlation length is defined using the correlation
among the normals of the system,
g11 g22 c
g11 g22 c
K −μ
=− 2
=− 2 ,
K +μ
g22 c
g11 c
We have performed Monte Carlo (MC) simulations on
triangular lattices of different sizes using a standard Metropolis
algorithm. We have performed simulations for several months
both on CPUs (Intel Core I7-3770) and GPUs (NVIDIA Tesla
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V. NUMERICAL RESULTS
A. Crumpling transition
In Figs. 3 and 4 we show the behaviors of the specific heat and dRg2 /dκ as functions of κ. The onset of a
critical behavior is clear in both figures. To compute the
maximum of both observables, we have fitted each curve near
its maximum using least squares (for L < 96 we have also used
the spectral density method). Monte Carlo has been performed
on the raw data in order to compute the error bars, both in the
value of the maximum and in its position.
10000
(dR2g/dκ)(T, L)
C2070). While the MC sweep of each site on the lattice on
CPUs is entirely sequential, in GPUs we can parallelize the
simulation using a kind of checkerboard scheme with four
colors (see Fig. 1 and Appendix C for details). We have
obtained a gain factor around 5 times simulating on GPUs.
We have simulated lattice sizes in the 16 L 128 range,
using free-boundary conditions. The thermalization protocol
has been as follows: (1) we have always started from a flat
configuration, (2) we have discarded the first 106 Metropolis
sweeps, and (3) we have analyzed the remainder of the
run using a logarithmic binning check of several nonlocal
observables in the most challenging simulations, i.e., for
κ κc .
After thermalization, we have saved 105 configurations
with at least 104 Metropolis sweeps between each pair of
saved configurations. Via the computation of the integrated
autocorrelation times [35] we have checked that all our
measures are fully independent. To estimate the error bars
of our observables we always used a jackknife method with
20 bins.
For each lattice size, we have considered several values of
κ around κc , as well as κ values well inside the flat phase; in
general, we have simulated the range 0.75 κ 3.0.
L=128
L=96
L=64
L=48
L=32
L=24
L=16
1000
100
10
1
0.76 0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96
T
FIG. 4. Behavior of the κ derivative of the squared gyration
radius as a function of temperature for all simulated lattice sizes. The
solid black curves provide the least-squares fit of the points near the
maximum to a quadratic form as described in the text. We have used
pentagons (L = 16), rhombs (L = 24), inverted triangles (L = 32),
triangles (L = 48), circles (L = 64), squares (L = 96), and crosses
(L = 128) to mark the points in the figure.
In order to characterize the critical properties (order of the
phase transition, critical exponents, etc.), we first monitor the
behavior of the maximum of the specific heat (see Fig. 5).
By using data with L > 16, we find a clear divergence of this
maximum following a power law with a background term, as
stated in Eq. (20), with α/ν = 0.756(40) (χ 2 /d.o.f. = 0.78/3,
where d.o.f. means the number of degrees of freedom of the
fit). This α/ν = 0.756(40) value is definitively different from
2, characteristic of a strong first-order phase transition, or even
from 1, which would indicate a weak first-order transition [36].
By using hyperscaling in two dimensions [α = 2(1 − ν)], we
can compute simultaneously both α and ν, to get
ν = 0.73(1),
6
5
CV(T, L)
4
3
L=128
L=96
L=64
L=48
L=32
L=24
L=16
α = 0.55(2) .
(27)
Using the ν exponent obtained from the scaling of the
maximum of the specific heat, we can study the κ position of
5.5
5
2
4.5
1
0.74 0.76 0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96
T
FIG. 3. Behavior of the specific heat as a function of temperature
for all simulated lattice sizes. The solid black curves provide the
least-squares fit of the points near the maximum to a quadratic form as
described in the text. We have used pentagons (L = 16), rhombs (L =
24), inverted triangles (L = 32), triangles (L = 48), circles (L = 64),
squares (L = 96), and crosses (L = 128) to mark the points in the
figure.
Cmax
4
3.5
3
2.5
2
1.5
0
20
40
60
80
100
L
120
140
FIG. 5. Scaling of the maximum of the specific heat (pentagons)
as a function of the size of the system. The solid line is a fit to Eq. (20).
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PHYSICAL REVIEW E 93, 022111 (2016)
10000
2
Peak dR g/dκ
Peak C
Fit
0.2
(dR g/dκ)max
0.1
100
2
κ-κc
1000
0.15
0.05
10
0
0.01
0.02
0.015
L-1/ν
0.025
1
FIG. 6. Position of the maximum of the specific heat (lower data,
see legend, pentagons) and of the maximum of dRg2 /dκ (inverted
triangles) as a function of L−1/ν . In this plot we have employed
1/ν = 1/0.73, as obtained from the fit of the peak of the specific
heat. We have used the position of the specific heat maximum to
obtain κc . Finally, we have fitted dRg2 /dκ by imposing the previous
obtained values of ν and κ; hence, there is a single free parameter
in the fit. The solid (specific heat) and dashed (gyration radius) lines
are “linear” fits in this scale. Notice that the gyration radius presents
stronger scaling corrections than the specific heat.
this maximum (see Fig. 6), denoted as κ(L). This κ(L) follows
the standard scaling equation (neglecting scaling corrections),
κ(L) = κc + AL−1/ν ,
(28)
where κc is the infinite volume critical coupling. By using the
ν value from Eq. (27), we obtain
κc = 0.773(1),
(29)
where we have taken into account only L > 32 data
(χ 2 /d.o.f. = 2.016/2). We have been unable to compute the
correction-to-scaling exponent of the crumpling transition.
We can redo the previous analysis on the derivate of the
squared gyration radius (Fig. 6). The scaling of the maximum
of dRg2 /dκ (see Fig. 7) provides us with the following
combination of exponents [see Eq. (25)]:
2νF +
1
= 2.86(1),
ν
(30)
with χ 2 /d.o.f. = 3.5/5 (L 16). Using again the ν value
obtained from the analysis of the specific heat, we can obtain
the Flory exponent, equivalently the Hausdorff dimension, of
the surface at criticality, taking into account the error bars in
our value of ν:
νF = 0.74(1),
dH = 2.70(2).
16
24
32
64
98 124
L
FIG. 7. Scaling of the maximum of the κ derivative (squares) of
the squared gyration radius, dRg2 /dκ, as a function of system size.
The solid line is a fit to Eq. (25).
surface normals. Moreover, this phase is known to host auxetic
behavior [1].
As mentioned in the Introduction, auxetic materials have
acquired huge importance from both the fundamental and
technological points of view [16]. The flat phase of CM
membranes unambiguously shows a negative value of the
Poisson ratio. However, a detailed study of the scaling
corrections in this phase is still lacking, while numerical data
should be extrapolated to infinite volume in order to perform
a proper comparison with analytical results. In particular, the
value σ = −1/3, obtained by the SCSA approximation [18],
has been hypothesized to characterize a unique universality
class for fixed-connectivity membranes [19].
We have first explicitly checked the isotropy of finite-size
membranes in the flat phase. To do this, we have monitored the
two different definitions of the Poisson ratio, namely, Eq. (26)
using g11 in the denominator or using g22 . Figures 8 and 9 show
σ
0.005
σ
0
(31)
-0.20
-0.22
-0.24
-0.26
-0.28
-0.30
-0.32
-0.20
-0.22
-0.24
-0.26
-0.28
-0.30
-0.32
κ=2.0, g22
κ=2.0, g11
κ=1.1, g22
κ=1.1, g11
0
20
40
60
80
100
120
140
L
B. Poisson ratio in the flat phase
We next study the low-temperature (equivalently, large κ)
flat phase of phantom crystalline membranes in more detail
(see Ref. [37] for a detailed computation of the η exponent). In
this ordered phase, long-range order exists in the orientation of
FIG. 8. Comparison between the computations of the Poisson
ratio using g11 and g22 as the denominator in Eq. (26); see legend.
Notice the strongly anisotropic behavior of the Poisson ratio for κ =
1.1 (bottom panel). For κ = 2 (top panel), isotropy holds for L > 32.
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PHYSICAL REVIEW E 93, 022111 (2016)
-0.2
-0.2
κ=3, g11
κ=3, g22
-0.22
Fit κ=3.0
κ=3.0
Fit κ=2.0
κ=2.0
-0.22
-0.24
-0.24
σ
σ
-0.26
-0.26
-0.28
-0.28
-0.3
-0.3
-0.32
-0.34
-0.32
0
0
20
40
60
80
100
120
L
140
FIG. 9. Comparison between the computations of the Poisson
ratio using g11 (squares) and g22 (crosses) as the denominator in
Eq. (26) for κ = 3; see legend. Isotropy holds for L 24.
the values of σ as obtained using these two definitions for three
different conditions on κ within the flat phase. For κ = 1.1,
which is near the crumpling transition, we have found a lack
of isotropy for almost all simulated lattice sizes (see Fig. 8).
However, we can safely assume isotropy for κ = 2 and L > 32
and for L > 16 if κ = 3.0 (see Figs. 8 and 9). In the following,
we will consider only κ = 2,3 and values of the lattice size for
which isotropy holds.
In order to extract the asymptotic value of the Poisson ratio,
we will use the standard scaling form,
A
,
(32)
Lω
where ω corresponds to the leading correction-to-scaling
exponent in the flat phase. We recall that the analytical
prediction is σ∞ = −1/3 [18].
We have obtained very good fits for both values of κ =
2,3. Specifically, for κ = 2 we obtain σ∞ = −0.30(1) and
ω = 1.3(9) (χ 2 /d.o.f. = 2.30/2), while κ = 3 leads to σ∞ =
−0.31(2) and ω = 0.76(47) (χ 2 /d.o.f. = 4.8/3). In both cases
the asymptotic value of the Poisson ratio is fully compatible
with the analytical prediction. We can try to improve our
analysis by fixing σ∞ = −1/3. For κ = 2.0 this leads to
ω = 0.44(6) (χ 2 /d.o.f. = 4.8/3), while for κ = 3 we obtain
ω = 0.46(5) (χ 2 /d.o.f. = 4.4/4).
Finally, we have tried a simultaneous fit of the κ = 2 and 3
data, i.e., a fit in which we assume that the values of σ∞ and
c in Eq. (32) are the same for both κ and in which we allow
for different values of A. The result of this joint fit is σ∞ =
−0.317(12) and ω = 0.63(20) (χ 2 /d.o.f. = 8.4/7). Further,
a joint fit in which we fix σ∞ = −1/3 leads to ω = 0.46(2)
(χ 2 /d.o.f. = 9.6/8). Figure√
10 displays the linear dependence
of the Poisson ratio with 1/ L.
σ (L) = σ∞ +
VI. DISCUSSION AND CONCLUSIONS
We have studied in detail important universal properties
associated with the flat phase of crystalline membranes,
specifically the nature of the crumpling transition and auxetics.
With respect to the former, we have found clear signatures
0.05
0.1
0.15
0.2
L-1/2
0.25
FIG. 10. Finite-size effects in the Poisson ratio for κ = 2 and
3. We have fixed the scaling correction exponent to 1/2 and the
asymptotic value to the analytical value, σ∞ = −1/3. We have used
crosses and a solid line fit (κ = 2) and squares and a dashed line fit
(κ = 2).
for a second-order phase transition. The values we obtain for
the critical exponents rule out completely an interpretation in
terms of a strong, or even a weak, first-order phase transition,
as had been proposed in the literature recently.
In order to assess the crumpling transition as a continuous
one, we have studied the critical behavior of the specific heat
and of the κ derivative of the gyration radius. In our numerical
analysis, we have employed realistic boundary conditions
for the membrane (free boundary conditions). We have also
avoided restricting the computation of our observables to an
inner region far from the boundary, as done elsewhere (see,
e.g., Ref. [2]).
The values we obtain for the critical exponents [α =
0.55(2), ν = 0.73(1), νF = 0.74(1), and dH = 2.70(2)] compare very well with previous results. For the specific-heat
exponent, we can mention α = 0.5(1) [2], 0.58(10) [38],
and 0.44(5) [39]. For the correlation length exponent, ν =
0.72(2) [39], 0.68(10) [38], and 0.85(14) [34]. Regarding the
value of the critical coupling, we can quote κc = 0.814(2) [38],
to be compared with our result, κc = 0.773(1). Finally,
the value for the Flory exponent, or, equivalently, for the
Haussdorf dimension, should be compared with the analytical
results νF = 0.73, dH = 2.73 [18] and those from numerical
simulations, νF = 0.71(3), dH = 2.77(10) [34].
Regarding a comparison with previous results, we believe
that in many cases the error bars provided for the critical
exponents reported in the literature have been clearly underestimated. We have computed the error bars after a careful study
of the different integrated correlation times, having simulated
among the largest available lattices. In spite of this, the value
of the critical coupling reported by some previous numerical
works is incompatible with our numerical result, which points
out a potential inadequacy of the corresponding statistical
analysis of numerical data. For instance, κc 0.82 [34]
and 0.814(2) [38] are previous determinations of the critical
coupling which are 18 standard deviations from the value
we obtain.
022111-6
UNIVERSAL BEHAVIOR OF CRYSTALLINE MEMBRANES: . . .
Beyond the nature and critical exponents of the crumpling
transition, we have further studied the value of the Poisson ratio
in the flat phase as a further universal property of CMs. This
seems particularly interesting in view of the huge scientific and
technological importance that auxetic materials have displayed
in recent years [16].
In particular, we have simulated three different values of κ
above the crumpling transition and in the flat phase in order
to compute the asymptotic value of the Poisson ratio. We have
discarded the smallest κ value due to the huge anisotropies that
occur for almost all simulated lattice sizes. By using the largest
two values of κ, we have found a very precise infinite-volume
extrapolation for the Poisson ratio which is in very good
agreement with the analytical value σ∞ = −1/3 computed in
Ref. [18]. In order to reach this conclusion, control of scaling
corrections has been required. We have specifically found
√
numerically that the scaling corrections behave as 1/ L.
Our extrapolated value for σ is compatible with previous
numerical work [2,15,19] which postulates such a value as
characteristic of a universality class of membranes with fixed
connectivity.
In principle, graphene is believed to provide a conspicuous
experimental realization of crystalline membranes. In this
context, the model provided by Eq. (2) is being intensively
employed on a phenomenological basis as a prototype to
describe the statistical-mechanical properties of this important
material system [5,6]. Nevertheless, some of the behaviors
of graphene, for instance, the temperature dependence of
the bending rigidity, seem to remain beyond this type of
phenomenological approach [5,6]. The fact that the Poisson
ratio of graphene is positive unless, e.g., temperature is
sufficiently high [40] or defects are introduced into the
crystalline structure [17] suggests that predicting realistic
values of σ also remains beyond current phenomenological
models of graphene membranes. From the theoretical point
of view, it will be interesting to identify what the nature of
the modifications to be made on generic models like Eq. (2)
is so that they can eventually improve upon this type of
prediction.
ACKNOWLEDGMENTS
This work was partially supported by Ministerio de
Economı́a y Competitividad (Spain) through Grants No.
FIS2012-38866-C05-01 and No. FIS2013-42840-P, by Junta
de Extremadura (Spain) through Grant No. GRU10158 (partially funded by FEDER), and by the European Union through
Grant No. PIRSES-GA-2011-295302. We also made use of
the computing facilities of Extremadura Research Centre
for Advanced Technologies (CETA-CIEMAT), funded by the
European Regional Development Fund (ERDF).
PHYSICAL REVIEW E 93, 022111 (2016)
as
1
1 αβ
αβ
HE (uα,β ) = d x μ uαβ − δαβ u − δ
2
2
1
β
(A1)
+ Kuαα uβ ,
2
2
where K = λ + μ is the compressibility modulus. From this
equation we can compute the stress tensor σαβ [42],
δHE
= Kδαβ uγγ (x)
σαβ (x) =
αβ
δu (x)
1 γ + 2μ uαβ (x) − δαβ uγ (x) .
(A2)
2
γ
(A3)
The Poisson ratio can be written as [43]
σ =−
K −μ
u22 =
,
u11 K +μ
(A4)
where the directions x 1 and x 2 on the substrate (the x plane)
on which the flat surface lives are denoted by the indices 1 and
2, respectively.
Using the linear response theorem, we can write [15]
uαβ (x)uγ δ ( y)c =
δuαβ (x)
,
δσ γ δ ( y)
(A5)
where (· · · )c denotes the connected average as elsewhere in
this work. In addition, taking the derivative of Eq. (A3), one
can obtain
K −μ
δuαβ (x)
= −
δαβ δγ δ
γ
δ
δσ ( y)
4μK
1
(δαγ δβδ + δαδ δβγ ) δ(x − y) . (A6)
+
4μ
Notice that, in a continuous system with a finite area, δ(0) =
1 [44]. Using the last two equations, we can write
K −μ
,
4μK
1
,
u12 (x)u12 (x)c =
4μ
2
K +μ
.
u11 (x) c = u222 (x) c =
4μK
u11 (x)u22 (x)c =
(A7)
(A8)
(A9)
Hence, we can finally write Eq. (A4) as
σ =
APPENDIX A: SOME ELASTICITY EQUATIONS
In this Appendix we recall some standard elasticity equations and use them to obtain Eq. (26), which has been employed
in this work to compute numerically the Poisson ratio.
The starting point is Eq. (7), considered as a Hamiltonian (see Ref. [15,41]). This equation can be rewritten
γ
Using σγ = 2Kuγ , it is possible to invert the last equation,
obtaining
1
1
1
γ
γ
σαβ (x) − δαβ σγ (x) .
uαβ (x) =
δαβ σγ (x) +
4K
2μ
2
K −μ
u11 u22 c
u11 u22 c
=− 2 =− 2 ,
K +μ
u11 c
u22 c
(A10)
which, when written in terms of the metric gαβ (x), gives us
Eq. (26) of the main text.
Note that we can compute the tangent vectors ∂i r as
differences. Here, i = 1,2,3 run on the three natural directions
of the triangular lattice, which are not orthogonal, while in the
022111-7
R. CUERNO et al.
PHYSICAL REVIEW E 93, 022111 (2016)
definition of σ we assume that the deformations are mutually
orthogonal [43]. Hence, we have defined two orthogonal
axes
√
x 1 and x 2 , taking x 1 = e1 and x 2 = (e2 + e3 )/ 3, where e1 ,
e2 , and e3 are the three natural unit vectors on the triangular
lattice. Finally, we compute the induced metric gαβ in this
basis.
APPENDIX B: RG COMPUTATION OF THE
CORRECTION-TO-SCALING EXPONENT
FOR THE FLAT PHASE
We can obtain a prediction within the RG framework for the
value of the correction-to-scaling exponent for the flat phase.
Following Refs. [1,41], within an expansion the renormalized
(dimensionless) elastic constants, μ̂ and λ̂, have the following
β functions (denoted as βμ̂ and βλ̂ ):
Setting = 2, we obtain λ1 −1.90 and λ2 −1.80. Hence,
ω −1.80 at this perturbative order.
Finally, we can compute the value of the Poisson ratio by
using the fact that, at the physical fixed point, λ̂/μ̂ = −1/3
and thus σ = λ̂/(λ̂ + 2μ̂) = −1/5. Notice that both values for
ω and σ are far from the corresponding MC results. We can
conclude that, due to the large value of , it is very difficult
to extract reliable predictions from the RG at this perturbative
order.
APPENDIX C: IMPLEMENTATION OF THE
CODE IN GPUS
(B4)
As widely acknowledged [45], pretty good gain factors
can be obtained by using graphic processing units (GPUs).
In our case, the gain was almost guaranteed, given that we
use mostly floating point variables that can be updated in
parallel following some kind of checkerboard algorithm. The
gain factor depends strongly on the system size, but we obtain
at least a factor of 5 times.
The compute unified device architecture (CUDA) implementation of the Metropolis algorithm stores a single copy of
the surface in the GPU memory. By operating only in the GPU
memory, we reduce memory controller load and reduce the
computation time required per iteration. The surface update
kernel is executed sequentially over a quarter of the surface
nodes and returns to CPU mode once all nodes have been
processed (see Fig. 1).
The structure of the algorithm requires performing an
atomic operation over a single variable to sum the total number
of updated node positions. This is the hardest bottleneck in the
algorithm.
Due to the parallel update structure, we need to generate
a random seed for each node in the surface, as opposed to
the CPU sequential version, in which a single seed is enough.
The number of blocks and threads per block in the CUDA
simulation is given by the surface size and the maximum
number of GPU registers assigned to each thread. The limit in
the number of registers that a kernel can use has a great impact
on the overall performance.
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d λ̂
= − μ̂ +
d
+
20A
,
c
d log l
8π 2 3
1 1
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= − λ̂ +
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(65 − 125π 2 )
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λ1 =
(B3)
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PHYSICAL REVIEW E 93, 022111 (2016)
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022111-9
γ
γ
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One way to obtain this equation
is to write a Schwinger-Dyson
2
αβ
equation using the identity d[uαβ ] δuδαβ (e−HE + d suαβ σ ) = 0,
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